If the divergence is zero at a point, that means that such point does not contribute with the field as source nor a sink. Is that right?

So, the divergence of a point measures how that point contributes as a source or a sink with the field?

The surface integral in the equation above means a certain area, right? Is that area the area of the entire surface (like a gaussian surface in the gauss's law) or the area of the micro-surface that is "around" the point I'm measuring the divergence on?

Usually I like to think in the dimensions of the conceps (units). I noticed that the unit of divergence will always be area/volume (m^-1). Does that have any meaning?

If someone can help me with some of these questions I would be grateful...